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Topological entropy is a common measure of the rate of mixing in a flow. It can be computed by partition methods, or by estimating the growth rate of material lines or other material elements. This requires detailed knowledge of the…
In many applications, the two-dimensional trajectories of fluid particles are available, but little is known about the underlying flow. Oceanic floats are a clear example. To extract quantitative information from such data, one can measure…
Deformation of material lines drives transport and dissipation in many industrial and natural flows. Here we report an exact Eulerian formula for the stretching rate of a material line, also known as the topological entropy, in a prototype…
Topological entropy serves as a viable candidate for quantifying mixing and complexity of a highly chaotic system. Particularly in turbulence, this is determined as the exponential stretching rate of a fluid material line that typically…
A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle…
Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the…
Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a…
We present a method for computing the topological entropy of one-dimensional maps. As an approximation scheme, the algorithm converges rapidly and provides both upper and lower bounds.
Stirring a two-dimensional viscous fluid with rods is often an effective way to mix. The topological features of periodic rod motions give a lower bound on the topological entropy of the induced flow map, since material lines must `catch'…
Topological entropy or spatial entropy is a way to measure the complexity of shift spaces. This study investigates the relationships between the spatial entropy and the various periodic entropies which are computed by skew-coordinated…
A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is…
Many dynamical phenomena in complex systems concern spreading that plays out on top of networks with changing architecture over time -- commonly known as temporal networks. A complex system's proneness to facilitate spreading phenomena,…
We study the evolution of turbulent magnetic fields from a topological point of view, invoking commonplace mathematical tools from general topology and dynamical systems theory which connect magnetic field evolution to time reversal…
In this paper, we present a detailed framework to analyze the evolution of the random topology of a time-varying wireless network via the information theoretic notion of entropy rate. We consider a propagation channel varying over time with…
Stirring of fluid with moving rods is necessary in many practical applications to achieve homogeneity. These rods are topological obstacles that force stretching of fluid elements. The resulting stretching and folding is commonly observed…
We compute the entanglement entropy in a composite system separated by a finitely ramified boundary with the structure of a self-similar lattice graph. We derive the entropy as a function of the decimation factor which determines the…
The curvature field is measured from tracer particle trajectories in a two-dimensional fluid flow that exhibits spatiotemporal chaos, and is used to extract the hyperbolic and elliptic points of the flow. These special points are pinned to…
We elucidate the topological features of the entanglement entropy of a region in two dimensional quantum systems in a topological phase with a finite correlation length $\xi$. Firstly, we suggest that simpler reduced quantities, related to…
Polymers in nonuniform flows undergo strong deformation, which in the presence of persistent stretching can result in the coil-stretch transition. This phenomenon has been characterized by using the formalism of nonequilibrium statistical…
In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the…